The graphics display the behavior under iteration of rational maps of the form:
f(z)=za+c+λ/zb
If we fix λ and color for convergence at all points z in the complex plane, we see the dynamical-plane, home of the Julia sets. If we fix the value of z and color for convergence at all points λ in the plane, we see the parameter plane, home of the Mandelbrot sets.
Dynamical Plane
Parameter Plane
λ= a + i.b
c = a + i.b
Degree: a
b
f(z)=z2+c+λ/z2
z=0 - λ-plane is mapped
Iterations Scale
OffsetX OffsetY
Show image center
See the Julia Set
Graphics: Click on the canvas to center at that point. Scroll to zoom in or out.
Numbers: Scroll over a value to increase or decrease it. Click on a label to reset the value.
Places of interest...
Click and hover below to change λ Click again to freeze λ Scroll to zoom
Right-click on the Image to save it.
What: A page illustrating "Singular Perturbations of Complex Polynomials" by Dr. Robert L. Devaney Bulletin (new series) of the American Mathematical Society, vol 50, num 3, July 2013, pages 391-429.
Why: The article was an inspiration to one of its readers.
Because: Lycophron has tangential attachement to the AMS.
How: The page relies on WebGL to push the computations to the graphics card.
Why How: A pure javascript implementation is excruciatingly slow. WebGl is fun. Scroll over a number to see the magic at work (e.g. Douady's Rabbitt, scroll over λ a).
What are: The places of interest are simply fixed sets of coordinates for the equations and a choice of color. Added are simple objects whose birth near a century ago would indicate the road to the dynamical plane. Most of the places listed are the coordinates for the figures that illustrate the Devaney article.
Tech: The limitations of this page are mainly inherited from the limitations of the graphics card displaying it. It should be noted however that each browser implements WebGL independently, and the results vary. All browsers tested (Chrome, Safari, Opera and Firefox on MacOSX) functioned fairly well. The best user experience occurred on Safari.
